Solar Potential in Saudi Arabia for Southward-Inclined Flat-Plate Surfaces

: The major objective of the present work is to investigate into the appropriate tilt angles of south-oriented solar panels in Saudi Arabia for maximum performance. This is done with the estimation of the annual energy sums received on surfaces with tilt angles in the range 15 ◦ –55 ◦ inclined to south at 82 locations covering all Saudi Arabia. The analysis shows that tilt angles of 20 ◦ , 25 ◦ and 30 ◦ towards south are the optimum ones depending on site. These optimum tilt angles deﬁne three distinct solar energy zones in Saudi Arabia. The variation of the energy sums in each energy zone on annual, seasonal and monthly basis is given; the analysis provides regression equations for the energy sums as function of time in each case. Furthermore, the spatial distribution of the annual global inclined solar energy in Saudi Arabia is shown in a solar map specially derived. The annual energy sums are found to vary between 1612 kWhm − 2 year − 1 and 2977 kWhm − 2 year − 1 across the country. Finally, the notion of a correction factor is introduced, deﬁned, and employed. This factor can be used to correct energy values estimated by a reference ground albedo to those based on near-real ground albedo.


Introduction
Installations with tilted solar collectors for exploiting the renewable energy source of the Sun have long been available in the market as commercial products. Solar flatplate panels are nowadays widely used for converting solar energy into electricity (PV installations) or hot water (solar thermosyphons, solar heating systems). These stationary systems consist of solar panels that receive solar radiation at fixed tilt angles with a southward orientation in the northern hemisphere.
For exploiting the available solar energy at a location or a region, prior knowledge of this potential is necessary. This is gained by solar radiation measurements on both inclined and mostly horizontal surfaces. Unfortunately, solar radiation measurements on horizontal planes are not common worldwide due to the purchase and maintenance costs of the radiometers required [1]. On the other hand, solar radiation measurements on inclined surfaces are really scarce [2]. To fill this gap, solar radiation models have been developed since the second half of the 20th century. A good account of the most widely used solar radiation models is given in [3].
Internationally, there have been many studies to estimate the performance of static solar systems. The main effort in such studies has been focused on finding the most suitable tilt angle(s) and orientation(s) throughout the year at various locations in the northern hemisphere for maximum retrieval of solar energy by flat-plate collectors in a year e.g., [4][5][6][7][8][9][10][11][12][13][14][15], or within seasons [16]. On the other hand, there are works on the technical improvement of solar panels for higher efficiency e.g., [17][18][19][20][21]. Nevertheless, the scarcity of solar radiation measurements has triggered studies to use solar radiation modelling e.g., [10,11,22] in order to derive the optimum tilt angle and orientation for maximum solar energy on solar flat-plate collectors. Other methods use a combination of groundbased solar data and modelling e.g., [2] or utilise solar data from international data bases e.g., [23,24]. Recently, a new method was presented by Kambezidis and Psiloglou [25] for Greece that estimates the optimum tilt angle of an inclined flat-plate collector with southern orientation for maximum solar energy gain. This method is followed in the present study.
As far as Saudi Arabia is concerned, some research has already appeared in the literature related to the present study. El-Sebaii et al. [26] estimated the global, direct, and diffuse solar radiation components on horizontal and tilted surfaces at Jeddah. Kaddoura et al. [23] estimated the optimal tilt angle for maximum energy reception by PV installations at 7 locations (cities) of the country. They suggested that the PV panels should be adjusted 6 times in a year for maximum performance, an outcome that burdens installation and maintenance costs because of the moving parts involved. Zell et al. [27] performed solar radiation measurements at 30 stations in Saudi Arabia during the period October 2013-September 2014 (1 year) to assess the solar radiation resource at these locations. The World Bank [28] produced the Global Solar Atlas, which includes Saudi Arabia. This solar map provides the distribution of the 3 solar radiation components over the country and is based on calculations in the period 1999-2018. Finally, Almasoud et al. [29] provided a study about the economics of solar energy in Saudi Arabia.
From the above it is clear that no attempt has been made so far to construct a solar map for Saudi Arabia to show the potential of flat planes inclined southwards for the exploitation of solar energy by appropriate systems. This gap is bridged by the present study, which includes three innovations. (i) For the first time, solar maps for Saudi Arabia for the maximum energy on optimally inclined flat surfaces towards south are derived. (ii) For the first time in Saudi Arabia three energy zones are identified for solar applications. (iii) The notion of the (ground-albedo) correction factor e.g., [30] is used and universal curves (nomograms) of this parameter in relation to the tilt angle and the ground-albedo ratio are derived for the first time worldwide to our knowledge.
The structure of the paper is as follows. Section 2 describes the data collection and data analysis. Section 3 deploys the results of the study, while Section 4 presents the conclusions and main achievements of the work. Acknowledgements and References follow.

Data Collection
Hourly values of the direct, H b,0 (in Wm −2 ), and diffuse, H d,0 (in Wm −2 ), horizontal solar radiation components were downloaded from the PV-Geographical Information System (PV-GIS) tool [31] using the latest Surface Solar Radiation Data Set-Heliostat (SARAH) 2005-2016 data base (12 years) [32,33]. This platform has been chosen for retrieving solar radiation data through a user-friendly tool that provides data for any location in Europe, Africa, Middle East including Saudi Arabia, central and southeast Asia and most parts of the Americas. Nevertheless, the platform provides solar maps for Europe, Africa, Turkey and central Asia only. The methods used by PV-GIS to calculate solar radiation from satellite are described in various works [30,34,35]. A set of 82 sites was arbitrarily chosen in order to cover the whole area of Saudi Arabia. Table 1 gives the names and the geographical coordinates of these sites, while Figure 1 shows their location in the map of the country. It should be noted here that the selection of the sites was based on the "inhabited" criterion (i.e., urban areas, 57 out of a total of 82); other 25 sites were added to the 57, but refer to uninhabited regions (sites with no names in Table 1). For this reason, the dispersion of the 82 sites is not uniform within Saudi Arabia. There should also be mentioned here that the downloaded hourly solar horizontal radiation values refer to those Appl. Sci. 2021, 11, 4101 3 of 24 from an unobstructed horizon (no effect of the ground on solar radiation) by not selecting the "calculated horizon" option in the PV-GIS tool. Table 1. The 82 sites arbitrarily selected over Saudi Arabia to cover the whole area of the country; ϕ is the geographical latitude, and λ the geographical longitude in the WGS84 geodetic system. The "unnamed" sites refer to those away from known locations.

Data Processing and Analysis
In step 1, the downloaded hourly data from the PV-GIS website were transferred from the coordinated universal time (UTC) into the Saudi Arabia local standard one (LST = UTC + 3 h). It must be mentioned here that the PV-GIS solar radiation values were provided at different UTC times for the various sites considered, e.g., at hh:48 or hh:09, where hh stands for any hour between 00 and 23. In step 2, the hourly global horizontal radiation, H g,0 , values were estimated as the sum H g,0 = H b,0 + H d,0 . In step 3, the original Sun's azimuth and elevation (SUNAE) routine introduced by Walraven [36], together with its modifications [37][38][39][40] renaming SUNAE to XRONOS (meaning time in Greek, X is pronounced CH), ran for the geographical coordinates of the 82 sites in the period 2005-2016 to derive the solar altitudes, γ; the XRONOS algorithm computed γ at the LST times so calculated in step 1. In step 4, all radiation and solar geometry values were assigned to the nearest LST hour (i.e., values at hh:48 LST or hh:09 LST were assigned to just hh:00 LST). That was done in order to have all values in the data base at integer hours. In the next step 5, only those hourly solar radiation values were retained for analysis that were greater than 0 Wm −2 , and for γ ≥ 5 • (to avoid the cosine effect). Also, the criterion of H d,0 ≤ H g,0 should be met at hourly level.  Table  1.

Data Processing and Analysis
In step 1, the downloaded hourly data from the PV-GIS website were transferred from the coordinated universal time (UTC) into the Saudi Arabia local standard one (LST = UTC + 3 h). It must be mentioned here that the PV-GIS solar radiation values were provided at different UTC times for the various sites considered, e.g., at hh:48 or hh:09, where hh stands for any hour between 00 and 23. In step 2, the hourly global horizontal radiation, Hg,0, values were estimated as the sum Hg,0 = Hb,0 + Hd,0. In step 3, the original Sun's azimuth and elevation (SUNAE) routine introduced by Walraven [36], together with its modifications [37][38][39][40] renaming SUNAE to XRONOS (meaning time in Greek, X is pronounced CH), ran for the geographical coordinates of the 82 sites in the period 2005-2016 to derive the solar altitudes, γ; the XRONOS algorithm computed γ at the LST times so calculated in step 1. In step 4, all radiation and solar geometry values were assigned to the nearest LST hour (i.e., values at hh:48 LST or hh:09 LST were assigned to just hh:00 LST). That was done in order to have all values in the data base at integer hours. In the next step 5, only those hourly solar radiation values were retained for analysis that were greater than 0 Wm −2 , and for γ ≥ 5° (to avoid the cosine effect). Also, the criterion of Hd,0 ≤ Hg,0 should be met at hourly level.
For estimating global solar irradiance on an inclined plane facing south, Hg,βS (in Wm −2 ), the isotropic model of Liu-Jordan [41] was adopted (β is the tilt angle of the inclined plane in respect to the local horizon, in degrees). The isotropic model was used to estimate the ground-reflected radiation from the surrounding surface, Hr,βS (in Wm −2 ), received on the inclined flat surface. This model has been adopted in the present study because it has proved as efficient in providing the tilted total solar radiation in many parts of the world as other more sophisticated models [42]. For a south-facing surface the received total solar radiation is given by [43]:  Table 1.
For estimating global solar irradiance on an inclined plane facing south, H g,βS (in Wm −2 ), the isotropic model of Liu-Jordan [41] was adopted (β is the tilt angle of the inclined plane in respect to the local horizon, in degrees). The isotropic model was used to estimate the ground-reflected radiation from the surrounding surface, H r,βS (in Wm −2 ), received on the inclined flat surface. This model has been adopted in the present study because it has proved as efficient in providing the tilted total solar radiation in many parts of the world as other more sophisticated models [42]. For a south-facing surface the received total solar radiation is given by [43]: where the subscript S denotes the south orientation of the inclined surface. According to Liu-Jordan [41]: H r,βS = H g,0 ·R r ·ρ g0 , (or ρ g ) cosθ = sinβ·cosγ·cos(ψ − ψ') + cosβ·sinγ.
where θ is the incidence angle (the angle formed by the normal to the inclined surface and the line joining the surface with the centre of the Sun), and ψ, ψ' are the solar azimuths of the Sun and of the inclined plane, respectively; the latter two solar parameters were estimated through the XRONOS algorithm, while ψ' was always equal to 180 degrees to south. The parameters R di and R r are called the isotropic sky-configuration and groundinclined plane-configuration factors, respectively. In the Liu-Jordan model the ground albedo usually takes the value of ρ g0 = 0.2 (Equation (3)). This value has also been used in the present study. Apart from using ρ g0 in the calculations, values of ρ g near to reality were also adopted and used here. To retrieve such values for the 82 sites, use of the Giovanni portal [44] was made; pixels of 0.5 • × 0.625 • spatial resolution were centered over each of the 82 sites for which monthly mean values of the ground albedo were downloaded in the period of the study (2005-2016); annual mean ρ g values were then computed and were used to re-calculate H g,βS .
In the present study, β varied in the range 15 • -55 • in increment of 5 • ; this range was chosen so that it fully covers all latitudes of Saudi Arabia (from ≈18 • N to 32 • N). For every site and tilt angle, hourly values of H g,βS were estimated twice from Equation (1); the first time the computations included ρ g0 = 0.2, and the second time the calculations were repeated with ρ g (=ground-albedo value from Giovanni). From the hourly H g,βS values, annual, seasonal and monthly solar energy sums (in kWhm −2 ) under all-sky conditions were estimated for all sites, all tilt angles and both ground-albedo values. Table 2 shows the maximum annual H g,opt. βS sums for all sites with their corresponding tilt angles, β. The derivation of those energy values was based on the optimum angle β (in the range 15 • -55 • in steps of 5 • ) for which maximum H g,βS was obtained. Along with each maximum H g,opt. βS the corresponding optimum β is shown. Differences of the annual energy sums, ∆H g,βS , were then derived through using ρ g and ρ g0 in Equation (3), i.e., ∆H g,βS = H g,βS/ρg . − H g,βS/ρg0 . Note that the optimum βs differ between the two cases (columns 2 and 3 in Table 2, respectively). The truth is that the reference value of ρ g0 is applicable to grassland areas; surfaces with different vegetation or no vegetation (e.g., tundras, deserts, snow-covered areas) may have reflectance far from 0.2 [42]. The differences ∆H g,βS are shown in Figure 2 for all 82 sites. From these Figs. it is seen that almost all differences are positive. This occurs for the sites having ρ g > ρ g0 , i.e., for a ground albedo greater than 0.2; the negative differences correspond to sites having ρ g < ρ g0 , i.e., for a ground albedo less than 0.2. It is considered that the H g,opt. βS/ρg values are closer to reality because of the use of the near-real ground-albedo values ρ g ; therefore, the present study deals with these "pragmatic" annual solar energy sums in the rest of the analysis. Figure 3 shows the distribution of ρ g over Saudi Arabia.

Annual Energy Sums and Solar Energy Zones
Considering the maximum energy sums and optimal tilt angles given in column 3 of Table 2 Table 2 for both ground-albedo cases of ρg and ρg0.  Table 2 for both ground-albedo cases of ρ g and ρ g0 .      mum β = 20° S, (ii) SEZ-B (orange circles) with optimum β = 25° S, and (iii) SEZ-C (green circles) with optimum β = 30° S. Contrary to this result, Zell et al. [27] divided the country into 5 geographical regions (central, eastern, southern, western, western inland) for the purpose of analysing the solar radiation data from [45]. Nevertheless, the division of Zell et al.'s did not obey any solar radiation criteria, while the SEZs in the present work really meet the mentioned energy criteria; therefore, it is of no practical value. The numbers in the symbols refer to the sites (column 1, Table 2).
Kaddoura et al. [23] estimated 12 optimal angles β for the 12 months of the year for Tabuk (#8 in Table 1), Al Jawf (#9), Riyadh (#10), Jeddah (#16), and Abha (#24). These angles were derived from modelling and they, therefore, have a purely theoretical value, since it is not practical at all to change the tilt angle of the solar panel frame every month. To the contrary, the present work gives a constant tilt angle throughout the year at any site in Saudi Arabia belonging to the same SEZ. Zell et al. [27] [27]. El-Sebaii et al. [26] agree with the optimum tilt angle for Jeddah as they find it to be 21.76 • , close to the 20 • of this study.

Monthly Energy Sums
The intra-annual variation of H g,opt. βS/ρg in each specific SEZ region is shown in Figure 5. Figure 5a refers to the sites in SEZ-A (optimal β = 20 • ), Figure 5b to the sites in SEZ-B (optimal β = 25 • ), and Figure 5c to the sites in SEZ-C (optimal β = 30 • ). The expressions for the lines that best-fit the means and their coefficient of determination, R 2 , are given in Table 3. It is seen that the R 2 statistic obtains high values; this allows a solar energy user or investor in Saudi Arabia to estimate the monthly energy production in any of the 3 SEZs in an accurate way by applying the regression equations. The graphs also contain curves for the mean ± 1 standard deviation (σ).  [23], respectively. The tilt angles from the present study (Table 2) are 25°, 30°, 25°, 20°, and 20°, respectively, which are very close to the average values of [27]. El-Sebaii et al. [26] agree with the optimum tilt angle for Jeddah as they find it to be 21.76°, close to the 20° of this study.

Monthly Energy Sums
The intra-annual variation of Hg,opt. βS/ρg in each specific SEZ region is shown in Figure  5. Figure 5a refers to the sites in SEZ-A (optimal β = 20°), Figure 5b to the sites in SEZ-B (optimal β = 25°), and Figure 5c to the sites in SEZ-C (optimal β = 30°). The expressions for the lines that best-fit the means and their coefficient of determination, R 2 , are given in Table 3. It is seen that the R 2 statistic obtains high values; this allows a solar energy user or investor in Saudi Arabia to estimate the monthly energy production in any of the 3 SEZs in an accurate way by applying the regression equations. The graphs also contain curves for the mean ± 1 standard deviation (σ).

Seasonal Energy Sums
Investors in solar energy installations are mostly interested in knowing the minimum and maximum possible energy received by their energy systems. This is interpreted as the solar potential during the winter and summer months. Therefore, this Section is devoted to analysing the solar energy totals during all seasons, i.e., spring (March-April-May), summer (June-July-August), autumn (September-October-November) and winter (December-January-February). Figure 6 presents the total solar energy received on a south-oriented flat surface in SEZ-A (Figure 6a), SEZ-B (Figure 6b), SEZ-C (Figure 6c), and all SEZs (Figure 6d) under all-sky conditions during spring, summer, autumn, and winter. The energy values are sums for each season averaged over all sites belonging to the same SEZ (or all SEZs). Table 3 gives the regression equations for the curves that best fit the mean ones in each case. It is interesting to observe that the fits are ideal (R 2 = 1) in all cases. Figure 7 shows the solar potential over Saudi Arabia in terms of annual H g,0 and H g,opt. βS/ρg sums. It is interesting to observe a gradual increase in the annual solar potential in the direction NE-SW for both horizontal and optimally inclined flat planes. Very similar results to the present study are given in the Solar Radiation Atlas for Saudi Arabia [45]. This gradient is due to two reasons. (i) Latitude: the higher the latitude, the less solar radiation is received on the surface of the Earth. (ii) Meteorology: more frequent precipitation is observed in the north-eastern part of the country, which is related to the precipitation occurring in southern Iraq and Iran [46].

Seasonal Energy Sums
Investors in solar energy installations are mostly interested in knowing the minimum and maximum possible energy received by their energy systems. This is interpreted as the solar potential during the winter and summer months. Therefore, this Section is devoted to analysing the solar energy totals during all seasons, i.e., spring (March-April-May), summer (June-July-August), autumn (September-October-November) and winter (December-January-February). Figure 6 presents the total solar energy received on a south-oriented flat surface in SEZ-A (Figure 6a), SEZ-B (Figure 6b), SEZ-C (Figure 6c), and all SEZs (Figure 6d) under all-sky conditions during spring, summer, autumn, and winter. The energy values are sums for each season averaged over all sites belonging to the same SEZ (or all SEZs). Table  3 gives the regression equations for the curves that best fit the mean ones in each case. It is interesting to observe that the fits are ideal (R 2 = 1) in all cases.  Figure 7 shows the solar potential over Saudi Arabia in terms of annual Hg,0 and Hg,opt.

Maps of Annual Energy Sums
βS/ρg sums. It is interesting to observe a gradual increase in the annual solar potential in the direction NE-SW for both horizontal and optimally inclined flat planes. Very similar results to the present study are given in the Solar Radiation Atlas for Saudi Arabia [45]. This gradient is due to two reasons. (i) Latitude: the higher the latitude, the less solar radiation is received on the surface of the Earth. (ii) Meteorology: more frequent precipitation is observed in the north-eastern part of the country, which is related to the precipitation occurring in southern Iraq and Iran [46].

Evaluation of the PV-GIS Tool
Given that all data used in this study were downloaded from the PV-GIS platform, the question arises as to the accuracy and reliability of these data. For this reason, various studies have presented validation results for the solar radiation PV-GIS-satellite-derived data by comparing them with ground-based solar radiation measurements from 30 BSRN stations [30,34,35]. The reported % differences in the form of "100*(PV-GIS-data-stationdata)/station-data" were found to vary between −14% to +11%. To further demonstrate this, comparison of monthly mean H g,0 values derived from solar radiation measurements continuously performed at the Actinometric Station of the National Observatory of Athens (ASNOA, 37. Figure 8 shows this comparison. Although there is an excellent agreement qualitatively (R 2 = 0.99), it seems that the PV-GIS-retrieved data overestimate the measured H g,0 by a factor of 1.1; this ratio is defined by any H g,0 value on the best-fit line to the corresponding one on the 1:1 line. In other words, the PV-GIS data are +10% higher than the ASNOA values, in accordance to the error range mentioned above. Nevertheless, as this factor is small and R 2 is high, the PV-GIS data are considered acceptable for solar energy analyses.

Evaluation of the PV-GIS Tool
Given that all data used in this study were downloaded from the PV-GIS platform, the question arises as to the accuracy and reliability of these data. For this reason, various studies have presented validation results for the solar radiation PV-GIS-satellite-derived data by comparing them with ground-based solar radiation measurements from 30 BSRN stations [30,34,35]. The reported % differences in the form of "100*(PV-GIS-data-stationdata)/station-data" were found to vary between −14% to +11%. To further demonstrate this, comparison of monthly mean Hg,0 values derived from solar radiation measurements continuously performed at the Actinometric Station of the National Observatory of Athens (ASNOA, 37.97° N, 23.72° E, 107 m above sea level) was made with respective values from the PV-GIS platform in the period 2005-2011. The ASNOA site was selected as Saudi Arabia does not possess any permanent solar radiation measuring network nor isolated measuring radiometric stations on a systematic basis. The only station that participated in many international networks (e.g., AERONET, BSRN) is the Solar Village one (18.23° N, 42.66° E, 2039 m above sea level) that was in operation during the period March 1998-December 2002, a period that does not include the starting date of PV-GIS (January 2005). Figure 8 shows this comparison. Although there is an excellent agreement qualitatively (R 2 = 0.99), it seems that the PV-GIS-retrieved data overestimate the measured Hg,0 by a factor of 1.1; this ratio is defined by any Hg,0 value on the best-fit line to the corresponding one on the 1:1 line. In other words, the PV-GIS data are +10% higher than the ASNOA values, in accordance to the error range mentioned above. Nevertheless, as this factor is small and R 2 is high, the PV-GIS data are considered acceptable for solar energy analyses.

Definition and Variation of the Correction Factor
This Section uses the notion of the correction factor, CF. This factor is defined by: CF = H g,opt. βS/ρg /H g,opt. βS/ρg0 . In other words, CF is the ratio of the annual H g,βS sum at each site of the 82, calculated twice, once for ρ g0 = 0.2 and a second time for ρ g = actual value; β is the tilt angle in the range 15 • -55 • . CF is named correction factor because it renders the maximum annual solar energy sum on a flat plane at a site in Saudi Arabia inclined to the south under the influence of the real terrain surrounding the location if this energy maximum value is known for an albedo value of 0.2. In other words, CF corrects the energy on an inclined surface under the influence of a ground albedo equal to 0.2 to that which is under the influence of near-real ground-albedo value. Figure 9 presents the variation of CF as function of β for all 82 sites; the controlling parameter is the ratio ρ r = ρ g /ρ g0 , which takes a different value at each site. It is seen that the higher the ρ r value is, the more concave the best-fit curve is (for CF > 1); to the contrary, the lower the ρ r value is, the more convex the best-fit curve becomes (for CF < 1). All the data points at every β correspond to the 82 sites. The distribution of ρ r at the 82 sites is shown in Figure 10.

Definition and Variation of the Correction Factor
This Section uses the notion of the correction factor, CF. This factor is defined by: CF = Hg,opt. βS/ρg/Hg,opt. βS/ρg0. In other words, CF is the ratio of the annual Hg,βS sum at each site of the 82, calculated twice, once for ρg0 = 0.2 and a second time for ρg = actual value; β is the tilt angle in the range 15°-55°. CF is named correction factor because it renders the maximum annual solar energy sum on a flat plane at a site in Saudi Arabia inclined to the south under the influence of the real terrain surrounding the location if this energy maximum value is known for an albedo value of 0.2. In other words, CF corrects the energy on an inclined surface under the influence of a ground albedo equal to 0.2 to that which is under the influence of near-real ground-albedo value. Figure 9 presents the variation of CF as function of β for all 82 sites; the controlling parameter is the ratio ρr = ρg/ρg0, which takes a different value at each site. It is seen that the higher the ρr value is, the more concave the best-fit curve is (for CF > 1); to the contrary, the lower the ρr value is, the more convex the best-fit curve becomes (for CF < 1). All the data points at every β correspond to the 82 sites. The distribution of ρr at the 82 sites is shown in Figure 10.  Figure 10a). The dotted lines are the best-fit curves to the data points of each site. The best-fit lines are expressed by 3 rdorder polynomials. For example, the upper best-fit curve is: CF = 2·10 −7 β 3 + 5·10 −6 β 2 + 1·10 −4 β + 0.9977 (R 2 = 1), while the lowest one: CF = −5·10 −8 β 3 − 2·10 −6 β 2 − 9·10 −5 β + 1.0006 (R 2 = 1). The blue horizontal line indicates CF = 1. Further, a diagram of CF vs. ρ r is shown in Figure 11, where a linear relationship is revealed and is shown for all sites. The controlling parameter in this case is β; as β increases, the slope of the linear fit to the data points decreases. The data points along each line correspond to the 82 sites.  Further, a diagram of CF vs. ρr is shown in Figure 11, where a linear relationship i revealed and is shown for all sites. The controlling parameter in this case is β; as β in creases, the slope of the linear fit to the data points decreases. The data points along each line correspond to the 82 sites. Despite the above, a more detailed analysis was performed for the function CF. In this analysis, the average values of CF were calculated for all βs and all sites belonging to the same SEZ, as well as all sites irrespective of SEZ. The results are shown in Figure 12 where the variation of CF with β in the range 15°-55° is shown. More particularly, th graphs give the variation of CF in a specific SEZ having the same optimum β, if the opti mum tilt angle includes all values in the above range, i.e., if all βs become optimum in specific SEZ. Table 4 gives the regression equations for the lines that best fit the data point in each SEZ and all SEZs too. These regression relationships have the same shape as thos in Figure 9. Despite the above, a more detailed analysis was performed for the function CF. In this analysis, the average values of CF were calculated for all βs and all sites belonging to the same SEZ, as well as all sites irrespective of SEZ. The results are shown in Figure 12, where the variation of CF with β in the range 15 • -55 • is shown. More particularly, the graphs give the variation of CF in a specific SEZ having the same optimum β, if the optimum tilt angle includes all values in the above range, i.e., if all βs become optimum in a specific SEZ. Table 4 gives the regression equations for the lines that best fit the data points in each SEZ and all SEZs too. These regression relationships have the same shape as those in Figure 9.  It is anticipated that the curves of CF vs. β ( Figure 9) and CF vs. ρr ( Figure 11) are universal (nomograms); this remains to be demonstrated at other locations in the world.

Conclusions and Discussion
The present study investigated the solar availability across Saudi Arabia on flat-plate solar panels with south orientation. The main objective was to find the optimum tilt angles of solar panels that produce maximum annual energy under all-sky conditions. This was achieved by calculating the annual energy sum on flat-plate surfaces with tilt angles in the range 5°-55° towards south with an increment of 5° at 82 sites across Saudi Arabia; the solar availability on a horizontal plane was also included. The calculations of the energy received on the tilted surfaces were repeated for both a ground albedo equal to 0.2 (reference value) and a near-real ground albedo.
The first outcome of the work was that the optimum tilt angles over Saudi Arabia are 20°, 25°, and 30° towards south. The second finding of the study was that these 3 optimum tilt angles group the 82 sites into 3 (solar energy) zones (SEZ), i.e., SEZ-A with 20° S, SEZ-B with 25° S, and SEZ-C with 30° S. The third conclusion was related with the variation of the annual maximum solar energy in each SEZ, i.e., 2192-2595 kWhm −2 year −1 (SEZ-A), 1612-2977 kWhm −2 year −1 (SEZ-B), 1702-2587 kWhm −2 year −1 (SEZ-C), and 1612-2977 kWhm −2 year −1 (all SEZs). Beside the annual energy sums, monthly solar energy ones averaged over all locations belonging to the same SEZ as well as to all SEZs were estimated It is anticipated that the curves of CF vs. β ( Figure 9) and CF vs. ρ r (Figure 11) are universal (nomograms); this remains to be demonstrated at other locations in the world.

Conclusions and Discussion
The present study investigated the solar availability across Saudi Arabia on flat-plate solar panels with south orientation. The main objective was to find the optimum tilt angles of solar panels that produce maximum annual energy under all-sky conditions. This was achieved by calculating the annual energy sum on flat-plate surfaces with tilt angles in the range 5 • -55 • towards south with an increment of 5 • at 82 sites across Saudi Arabia; the solar availability on a horizontal plane was also included. The calculations of the energy received on the tilted surfaces were repeated for both a ground albedo equal to 0.2 (reference value) and a near-real ground albedo.
The first outcome of the work was that the optimum tilt angles over Saudi Arabia are 20 • , 25 • , and 30 • towards south. The second finding of the study was that these 3 optimum tilt angles group the 82 sites into 3 (solar energy) zones (SEZ), i.e., SEZ-A with 20 • S, SEZ-B with 25 • S, and SEZ-C with 30 • S. The third conclusion was related with the variation of the annual maximum solar energy in each SEZ, i.e., 2192-2595 kWhm −2 year −1 (SEZ-A), 1612-2977 kWhm −2 year −1 (SEZ-B), 1702-2587 kWhm −2 year −1 (SEZ-C), and 1612-2977 kWhm −2 year −1 (all SEZs). Beside the annual energy sums, monthly solar energy ones averaged over all locations belonging to the same SEZ as well as to all SEZs were estimated under all-sky conditions. Regression equations were provided as best-fit curves to the monthly mean energy sums that estimate the solar energy potential per SEZ (and all SEZs) with great accuracy (R 2 ≥ 0.90). These expressions may prove very useful to architects, civil engineers, solar energy engineers, and solar energy-systems investors in order to assess the solar energy availability in Saudi Arabia throughout the year.
Seasonal solar energy sums were also calculated. They were averaged over all sites in the same SEZ as well over all sites (all SEZs) as in the case of the annual and monthly sums, under all-sky conditions. For every case, regression curves that best fit the mean values were estimated with greatest accuracy (R 2 = 1). Maximum sums were found in the summer (730 kWm −2 ), and minimum ones in the winter (450 kWm −2 ), as expected.
A correction factor, CF, was defined as the ratio of the maximum solar energy sum derived from calculations with near-real ground albedos, ρ g , to that derived with the reference value, ρ 0 . A graph of CF as function of the tilt angle (in the range 15 • -55 • ) showed an exponential growth for sites having ratios ρ g /ρ 0 > 1, or an exponential decay in the cases of ρ g /ρ 0 < 1. Such curves are assumed to be universal (nomograms), but this remains to be proved for other locations in the world with different climate and terrain characteristics. Graph of CF as function of ρ g /ρ 0 was prepared for different values of the tilt angle (in the range 15 • -55 • ). Best-fit lines to the data points were estimated and found to be linear of decreasing slope with decreasing tilt angle. These curves are also assumed to be nomograms, but this also has to be proved for other locations in the world with different climate and terrain characteristics.
Three innovations appeared in the present study: (i) For the first time, solar maps for Saudi Arabia of the maximum energy received on optimally-inclined flat surfaces towards south were derived; (ii) for the first time, three energy zones were identified in Saudi Arabia for solar applications; (iii) universal curves (nomograms) of CF in relation to the tilt angle and ρ g /ρ 0 were derived.
Based on the adopted methodology, some guidelines can be given here to interested solar energy scientists and/or solar energy entrepreneurs for applying them in their territory. If a solar radiation station exists in the area, hourly or daily values of the solar global horizontal radiation must be collected for a climatological period of 10 years at least. If no solar radiation exists, then data from a relative website (e.g., BSRN, GEBA, PV-GIS, ARM) can be obtained. In the extreme case that this option is not possible, use of a solar energy model can be made to derive the anticipated data from other available variables (e.g., meteorological parameters). Then, transposition of the selected data from horizontal to inclined planes towards the local south must take place by varying the tilt angles in a range that includes the geographical latitude of the site. In the next step, the annual solar energy sums on the inclined planes have to be calculated and the highest energy sum must be selected; then the corresponding tilt angle is the optimum for the site. The transposition can be achieved by selecting the wishful model (the L-J model is sufficient). It is recommended that a near-real ground albedo value is used in these calculations; if knowledge of this value is not available for the site, use of the nomogram of Figure 11 can be made to correct the solar energy sums by selecting the appropriate tilt angle. Finally, monthly solar energy sums can be estimated and regression lines be derived that can be used as guidelines for estimating the expected solar energy on flat planes with the selected tilt angle.
As far as the significance of the results of the present study is concerned in the solar industry and the society of Saudi Arabia, this can be summarized in the following. The solar industry has now a rule for the inclined parts of the solar system-support frames; the inclined surfaces of the supporting frames must have angles of 20 • , 25 • or 30 • depending on the SEZ to be installed. It is assumed that the orientation of the supporting frame has to be towards south. This rule may reduce installation costs because of the standardisation of the supporting frame. On the other hand, the society may get indirect benefits from governmental initiatives regarding policies that promote renewable energy sources in Saudi Arabia (and especially solar energy sources) because of the new knowledge gained through the present work. Data Availability Statement: The solar radiation data and the ground-albedo ones for Saudi Arabia are publicly available and were downloaded from the PV-GIS platform (https://ec.europa.eu/jrc/ en/pvgis) and the Giovanni website (https://giovanni.gsfc.nasa.gov/giovanni/), respectively. The ASNOA solar radiation data are available on request, but they were used on a self-evident permission to the second author as ex. member and now Emeritus Researcher in the Institution.